Module 4: Factorial Treatment Structure

Recall One-way Treatment Structures

One Factor at a Time

• Multiple levels -> \(t\) treatments
• Compare the mean response between the treatments

What are we missing?

• Effects may depend on context
• Optimal choice may change depending on that context
• One factor doesn’t always tell the full story

Example 4.1: Blood Pressure

Incidence of high blood pressure in the United States continues to rise, and optimal strategies for treatment remain an open question. Most physicians agree that effective treatment likely involves a combination of lifestyle changes and medication, rather than relying on a single approach.

Researchers plan to study middle-aged U.S. women with moderate hypertension who are otherwise healthy (i.e., eating a standard diet and not taking regular medications).

Goal: Identify optimal combinations of diet and medication for lowering and maintaining blood pressure as measured by the change in blood pressure (mmHg) for each participant.

Example 4.1: Blood Pressure

200 volunteers who meet the study inclusion criteria are recruited.

Study Design A (A common first attempt)

Study Design A (A common first attempt)

Researchers then find:

1. The Mediterranean diet lowers blood pressure, on average.
2. ACE inhibitors lower blood pressure more than other medications, on average.

Is the optimal strategy to recommend the Mediterranean diet and ACE inhibitors together?

Study Design B (A full factorial)

Each participant is randomly assigned to one combination of:

This results in t = _____ treatment combinations, with r = _____ participants per combination.

What can we learn from Design B that we can’t from Design A?

Full Factorial \((a \times b\times c \times ...)\)

Example: 2 diets x 4 medications

• _____ x _______ full factorial

Example (Extended): 2 diets x 4 medications x 2 exercise routines

• _____ x _______ x _______ full factorial

Treatment Structure for a Full Factorial

• Name:
• Factors:
• Levels:
• t = _____ with Treatments:

A 2 x 4 full factorial with factors Diet (2 levels - Standard American, Mediterranean) and Medication (4 levels - Diuretic, ACE inhibitor, Beta blocker, and Placebo) for a total of t = 8 treatment combinations (e.g., Standard American x Diuretic, Standard American x ACE, …, Mediterranean x Placebo).

Experimental Structure for a Full Factorial

• Name:
• Experimental units: r = _____
• Measurement unit:

Diet and Medication treatment combinations are assigned to middle-aged U.S. women with moderate hypertension who are otherwise healthy (e.u.) in a CRD with r = 25. The change in blood pressure (mmHg) is measured for each woman in the study (m.u.).

Create a Full Factorial CRD in R

library(edibble)
library(tidyverse)

des <- design(name = "BP 2 x 4 Full Facotrial CRD") |> 
  set_units(woman = 200) |>                  
  set_trts(Diet = c("Standard American", 
                    "Mediterranean"),
           Medication = c("Diuretic", 
                          "ACE", 
                          "Beta blocker", 
                          "Placebo")
           ) |> 
  allot_trts(Diet*Medication ~ woman) |>                
  assign_trts("random")  

factorial_table <- serve_table(des)
factorial_table$change_bp <- NA

head(factorial_table)

# An edibble: 6 x 4
     woman              Diet   Medication change_bp
  <U(200)>            <T(2)>       <T(4)>          
     <chr>             <chr>        <chr> <lgl>    
1 woman001 Standard American Beta blocker NA       
2 woman002     Mediterranean Beta blocker NA       
3 woman003     Mediterranean          ACE NA       
4 woman004     Mediterranean Beta blocker NA       
5 woman005 Standard American Beta blocker NA       
6 woman006 Standard American     Diuretic NA       

Create a Full Factorial CRD in JMP

DOE > Classical > Full Factorial